If you have a joint tenancy and your relationship with your joint tenant has broken down and he/she has left the home, you will probably not want his/her name on your rent book.
If your joint tenant agrees to have his/her name taken off the rent book, the local authority or housing association will do this. However, if he/she refuses to have his/her name removed, it is more complicated and you will have to provide evidence that the relationship has broken down.
If you are a partner left living alone in the family home, the local authority or housing association may consider that your house is too big for you and offer you alternative accommodation, unless you can argue that you have occasional custody of your children and need somewhere for them to stay.
Contact the Housing Department of your local authority or housing association.